Magnetized black holes:
ionized Keplerian disks and
ultra-high energy cosmic rays
Zdeněk Stuchlı́k
Silesian University in Opava, Czech Republic
in collaboration with M. Kološ, A. Tursunov, R. Pánis
Recent Progress in Relativistic Astrophysics
6-8 May 2019, Fudan University (Shanghai, China)
Magnetized black holes:
ionized Keplerian disks and ultra-high energy cosmic rays
Charged particle motion around magnetized BHs
Ionized Keplerian disks and their stability
Possibility of Ultra-High-Energy Cosmic Rays created by magnetic Penrose process
Properties of charged particle motion in the field of magnetized BHs imply four possible
regimes of behavior of ionized Keplerian disks: survival in regular epicyclic motion,
transformation into chaotic toroidal state, destruction due to fall into BH, destruction due to
escape to infinity along magnetic field lines (Kerr BH only). Due to extremely efficient
magnetic Penrose process particles escaping to infinity could form UHECR.
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Section 1
Charged particle motion around magnetized BHs
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Magnetized black holes
Kerr–Newman BHs
with internal electromagnetic field
regular motion of charged test particles - separability of EoM due to hidden
symmetry of Kerr–Newman background
magnetized BHs
in external electromagnetic fields
chaotic motion motion of charged particles
simplest Wald solution for asymptotically uniform magnetic field
induced charge
electromagnetic field created by electric flow in accretion disks
black hole immersed in a magnetar field
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Equations of motion - Hamiltonian formalism
1 αβ
1
g (πα − qAα )(πβ − qAβ ) + m2 ,
2
2
kinematical four-momentum pµ = muµ , generalized (canonical) π µ = pµ + qAµ
equations of motion - Hamilton equations, ζ = τ /m
H=
∂H
dxµ
≡ pµ =
,
dζ
∂πµ
dπµ
∂H
=− µ
dζ
∂x
Gravity gµν : axially symmetric spacetime (Kerr)
ds2 = gtt dt2 + 2gtφ dtdφ + gφφ dφ2 + grr dr2 + gθθ dθ2
Magnetic field Aα : external uniform mag. field B, electromagnetic 4-vector
At =
B
Q
Q
(gtφ + 2agtt ) − gtt − ,
2
2
2
Aφ =
B
Q
(gφφ + 2agtφ ) − gtφ
2
2
non-charged BH Q = 0 vs. BH (Wald) charge black generated by rotation
Schwarzschild BH - Aφ component only → no particle acceleration
Kerr BH - both Aφ & At are present → causing charged particle acceleration
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Reduction to the two-dimensional dynamics
symmetry => conserved quantities: energy E and angular momentum L
−E = πt = gtt pt + gtφ pφ + qAt ,
L = πφ = gφφ pφ + gφt pt + qAφ
two degree of freedom Hamiltonian, 4D phase space (r, θ; pr , pθ )
H=
1 rr 2 1 θθ 2 f
g pr + g pθ + HP (r, θ)
2
2
fP = 0)
condition E = Veff (r, θ) is energetic boundary for particle motion (H
Veff (r, θ) =
α = −g tt ,
−β +
p
β 2 − 4αγ
,
2α
β = 2[g tφ (L − q̃Aφ ) − g tt q̃At ],
γ = −g φφ (L − q̃Aφ )2 − g tt q̃ 2 A2t + 2g tφ q̃At (L − q̃Aφ ) − 1
B > 0 magnetic repulsion: ISCO strongly shifted inwards -can cross photon sphere
B < 0 magnetic attraction: ISCO slightly shifted inwards
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Section 2
Ionized Keplerian disks and their stability
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Ionization of particles from neutral accretion disk
ionization process: irradiation, decay,...
neutral particle (1) → two charged (2) + (3)
πα(1) = πα(2) + πα(3) ,
pα(1) = pα(2) + qAα + pα(3) − qAα
m(1) ≥ m(2) + m(3) ,
0 = q(2) + q(3)
very common situation
m(1) ∼ m(2) m(3) , → pα(1) = pα(2)
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Possible fates of ionized Keplerian disks
A) survival in regular epicyclic motion
B) transformation into chaotic toroidal state
C) destruction due to fall into BH
D) destruction due to escape to infinity along magnetic field lines (Kerr BH only)
Combined effect of gravitational en electromagnetic interactions governed by the
dimensionless ”magnetic intensity” parameter B
B=
qBGM
2mc4
B = 0.004
electron
10−5 Gs
proton
0.02 Gs
Fe+ charged dust
1 Gs
109 Gs
For stellar mass black hole M ≈ 10M , we can have one electron e- in the magnetic
field B = 10−5 Gs or charged dust grain (one electron lost, m = 2 × 10−16 kg) in field
B = 109 Gs - the absolute value of magnetic field parameter is the same in both cases
B = 0.004.
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Fate of ionized Keplerian disk as related to mag.field B
Disk is orbiting around Schwarzschild BH with inclination angle θ0 = 1.37
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Fate of ionized Keplerian disk as related to inclination θ0
Disk is orbiting around Schwarzschild BH with magnetic field parameter B = 0.1
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Determination of chaos in charged particle dynamic
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Determination of chaos in charged particle dynamic
15
1.6
10

B
1.1
r0
1.5
-15
-15
k
ut
dis
ne
tion
e
r
acc
-5
0.8
0.7
1.0
ral
-10
D2
equatorial plane
-10
1
θ
0
-5
1.4
1.3
5
z
2.0
θ0
0.5
0.4
0.2
0
5
10
15
0.5
0.1
6
6.5 7.2 7.8 8.4
9
9.6 10.2 10.8 11.4 12
r
x
For chaos determination and chaotic behaviour description, following methods have been
tested: Box-Counting, Correlation Dimension, Lyapunov Exponent, Recurrence Quantification
Analysis (RQA) with variants and Machine Learning with various algorithms.
• R. Pánis, M. Kološ, and Z. Stuchlı́k, Determination of chaotic behaviour in time series
generated by charged particle motion around magnetized Schwarzschild black holes, submitted
to Eur. Phys. J. C (2019)
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θ
Box-counting
Correlation dim.
Lyapunov exp.
RQA - RR
00:00:54
03:59:57
01:42:17
05:16:48
RQA - DET
RQA - ENTR
RQA - LL
ML - Random Forest
05:16:48
05:16:48
05:16:48
00:02:57
ML - Neural Network
ML - Linear Regression
ML - Nearest Neighbors
ML - Gradient Boosted Trees
00:02:08
00:03:24
00:02:11
00:02:13
r
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Section 3
Possibility of Ultra-High-Energy Cosmic Rays created by
magnetic Penrose process
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Chaotic scattering and magnetic Penrose process
Ionized neutral particle accelerated by the electric field generated by influence of BH
rotation on the magnetic field.
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Observation of Ultra-High-Energy Cosmic Rays
In UHECR there are particles with energies E > 1018 eV.
Few things we know: UHECR are charged particles, extremely rare, probably proton
dominated flux or iron nuclei, extra-Galactic origin
Energetic accelerator - mechanism is unknown: extra dimensions, new particles. . .
From supermassive black holes? Black hole thermodynamics: 29% of BH’s energy is
available for extraction (rotation), for extremely rotating SMBH of 109 solar mass the
available energy is 1074 eV
Let’s have one particle with rest mass/energy E0 it can
take energy from BH with efficiency η and got energy
E = E0 + ηE0
Penrose (1969) - astro. problematic η ≤ 0.2
Wagh et al. (1985) – astro. relevant η ∼ 1
electromagnetic version of Penrose process
Blandford & Znajek (1977) – BH energy extracted
by force-free plasma currents η ∼ 3
Many other versions with different efficiencies
But for UHECR energies one needs η > 108 !
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Particle ionization and magnetic Penrose process
rotating Kerr black hole - metrics with mixed component gtφ 6= 0
→ both electric field At and magnetic component Aφ for uniform magnetic field
→ BH can electromagnetically accelerated charged particles
particle with mass m and charge q, has kinematic four-momentum pµ = muµ ,
generalized four-momentum πµ = pµ + qAµ
Let neutral particle (1) decay into two charged particles (2) and (3)
πα(1) = πα(2) + πα(3) ,
m(1) ≥ m(2) + m(3) ,
0 = q(2) + q(3)
pα(1) = pα(2) + qAα + pα(3) − qAα
The term qAt can be arbitrary large → particle energy E can be changed dramatically
E = −πt = −gtα pα − qAt
(1)
→ 2nd particle with charge sign identical to the BH charge will be repelled
→ 3th particle with charge different to the BH will be captured by BH. BH will be discharged
and loses some energy.
• Z.Stuchlı́k, M.Kološ: Acceleration. . . , EPJc 76 (1), 1-21 (2016), [arXiv:1511.02936].
• A.Tursunov et al. Supermassive BH as a source of UHECR just submitted
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Particle ionization in vicinity of magnetized BH
1st neutral particle (black, neutron) has been located on accretion disk inner edge, and starts
to fall into BH. It will decay into two ionized particles 2nd (blue, proton) and 3rd (red,
electron). While the 3rd particle (red) fall inside BH with negative energy the second (blue)
got more energy and escape along magnetic field line.
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Particle ionization in vicinity of magnetized BH
1st neutral particle (black, neutron) has been located on accretion disk inner edge, and starts
to fall into BH. It will decay into two ionized particles 2nd (blue, proton) and 3rd (red,
electron). While the 3rd particle (red) fall inside BH with negative energy the second (blue)
got more energy and escape along magnetic field line.
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Efficiency of magnetic Penrose process
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Energy of accelerated protons
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Summary
Ionization of Keplerian disks can lead to
epicyclic oscillations (QPOs)
transmutations to toroidal structures
destruction of part of the disk
Around Kerr BH ionized particles can be accelerated with efficiency of the process
greater than 1010 so that protons can be accelerated to energy
1021 eV around supermassive BH with M = 109 M
and B = 104 Gauss
1016 eV around supermassive BH SgrA* with M = 4 · 106 M
and B = 10 Gauss
Thank you for your attention
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